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Numerical Modeling of the Near-subsurface
Temperature Distributions in Presence of Time Varying
Air Temperature in the Boundary Condition and Space
Varying Temperature for the Initial Condition Using
COMSOL Multiphysics
M. Ravi, D.V. Ramana, R.N. Singh
CSIR- NATIONAL GEOPHYSICAL RESEARCH INSTITUTE
HYDERABAD
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Objective
Develop a model to describe the distribution of
temperatures in the subsurface due to variations in the
surface air temperature.
Develop a numerical solution for which exponential
terms are included in both initial and robin type
boundary condition that can be apply to simulate for
future climates.
To observe the effect of interaction between the air and
the surface temperatures, groundwater velocity on the
subsurface thermal structure in changing climate.
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Introduction Heat is transported into the near subsurface
mainly by conduction and advection
Subsurface temperature distribution is
significant in thermo-physical,
climatological problems.
Analysis of Surface air temperatures gives
evidences for warming or cooling of the
near surface (Wigley et al,) thus perturbs
into the subsurface, develops an envelope
or signatures of thermal regime.Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Methodology Signatures of the climate change can be inferred by assuming one
dimensional transient heat transfer in porous media with groundwater velocity in the subsurface is given by
-density
-specific heat
u -vertical groundwater velocity
K -thermal conductivity of soil
Suffixes ‘s’ and ‘f’ refers to solid matrix and fluid
A(z,t) - Source/sink
),,()()(2
2
TtzAz
TK
z
TuC
t
TC fpsp
pC
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Initial condition:
A super-imposed linear and exponential function is used and is given by
T(x,0)=T0 + a*x + delt*exp(d*x)
Boundary condition:
Robin type boundary condition that relates surface heat flux due to both SAT and SST is used to access the atmospheric contribution to subsurface thermal profile and is given by
k =H*(T-(TA + b*exp(c*t)))t
T
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Parameters:Name Expression Description
T0 15[degC] Temperature of soil surface
TA16.5[degC] Air temperature on surface
k 2.5[W/(m*K)] thermal conductivity
K 6.1[m^2/yr] thermal diffusivity
u 0.5[m/yr] Mean annual vertical
groundwater velocity
H 0.2[W/(m^2*degC)] Heat transfer coefficient
a 0.022[degC/m] Temperature gradient
b 1.60[degC]
c 0.0116[/yr]
delt 2.0[degC]
d -0.02[/m]
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Solution of the problem Analytical solution for simple initial and boundary condition is
given by (Rajeev et al. 2012)
Initial condition:
T = T0 + aZ at t = 0 for all Z > 0
Boundary condition:
K*dT/dz= H(T −TA− bt) atZ=0 for all t > 0
Solved Temperature distribution with space and time in laplace domain.
T (z,t)= (T0 + aZ)/p− aU/p2+[K1 (T0 − TA) − a]exp (m2Z)/(p(m2 − K1))−(K1 (aU + b)/(p2(m2 − K1)))exp (m2Z)
where m2=U/(2k)-sqrt(U^2/(4*k^2 + p/k))
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
The above problem is
Solved numerically by
using COMSOL Multiphysics
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Finite element method using COMSOL
Multiphysics
• The problem is solved numerically by Finite Element Method using COMSOL Software.
• One dimensional heat conduction–advection equation in the subsurface is:
Number intervals are 2
Discretized for 200 elements
The solution is obtained by using iterative method
This iterative solver uses conjugate gradient method with error less than 0.001
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Results and discussions Temperature-depth profiles for annual
groundwater recharge (0.5[m/yr])
When the ground is recharging
with annual mean vertical ground-
water velocity, surface air(climate)
is forcing the earth’s surface to
warm.
Temperature profiles are observed
to be reversed after certain depth
where the effect of air and soil
interaction ceases.Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Temperature-depth profiles for annual
discharge (-0.5[m/yr]) Discharge will enhances the surface
to warm from the deeper subsurface
by advection.
When the ground is discharging the
temperature profiles will follows a
linearly decreasing trend from the
bottom of the subsurface due to
temperature gradient.
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Temperature-depth profile for annual
groundwater velocity (10^-5[m/yr])
It is observed that when the
ground is covered with ice and
the surface air with snow,
thermal profile follows an
exponential trend due to initial
condition and then it coincide
with the geothermal gradient.
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Temperature-depth profiles for different ‘H’ with no
flow, annual recharge and discharge rates
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Temperature-depth profile for various annual
groundwater velocities
Groundwater velocity is an
intensive parameter in
transporting heat in the
subsurface.
As annual groundwater velocity
increases, subsurface is warming
from the above with respect to
depth.
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Discussions
Temperature –depth profiles obtained by the
exponential model show that the recharge can
accelerate shallow subsurface warming, whereas
upward groundwater discharge can enhance deeper
subsurface warming.
With higher values of H, faster propagation of heat is
observed in the subsurface.
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Conclusions A numerical solution to the transient one-dimensional
conduction-advection equation was developed for air temperature projections.
These studies are useful to set the numerical results for the complex problems where it is not possible to get analytical solutions.
Sensitivity of ground recharge or discharge, interaction of surface air with surface soil on temperature profiles for subsurface are discussed.
These numerical solutions have potential use in understanding the future climate change by observing the thermal profiles.
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore
Thank You
Excerpt from the Proceedings of the 2014 COMSOL Conference in Bangalore